# Mechanics of allostery: contrasting the induced fit and population shift scenarios

## TL;DR

This study investigates the mechanical principles underlying allosteric regulation in proteins, contrasting induced fit and population shift models, and identifies optimal elastic properties that support cooperative binding and protein function.

## Contribution

The paper provides empirical evidence and a mechanical model analysis showing how elastic modes influence allosteric mechanisms and their robustness to mutations.

## Key findings

- Allosteric conformational change occurs along a soft elastic mode with high shear regions.
- Optimal stiffness for cooperative binding scales as 1/N, where N is the number of residues.
- Population shift scenario is more robust to mutations affecting stiffness.

## Abstract

In allosteric proteins, binding a ligand can affect function at a distant location, for example by changing the binding affinity of a substrate at the active site. The induced fit and population shift models, which differ by the assumed number of stable configurations, explain such cooperative binding from a thermodynamic viewpoint. Yet, understanding what mechanical principles constrain these models remains a challenge. Here we provide an empirical study on 34 proteins supporting the idea that allosteric conformational change generally occurs along a soft elastic mode presenting extended regions of high shear. We argue, based on a detailed analysis of how the energy profile along such a mode depends on binding, that in the induced fit scenario there is an optimal stiffness $k_a^*\sim 1/N$ for cooperative binding, where $N$ is the number of residues involved in the allosteric response. We find that the population shift scenario is more robust to mutation affecting stiffness, as binding becomes more and more cooperative with stiffness up to the same characteristic value $k_a^*$, beyond which cooperativity saturates instead of decaying. We confirm numerically these findings in a non-linear mechanical model. Dynamical considerations suggest that a stiffness of order $k_a^*$ is favorable in that scenario as well, supporting that for proper function proteins must evolve a functional elastic mode that is softer as their size increases. In consistency with this view, we find a significant anticorrelation between the stiffness of the allosteric response and protein size in our data set.